550.171 Discrete Mathematics  Spring 2005
Midterm #1
Name
1.
(a) [5 pt] State a careful de¯nition of what it means for a positive integer to be composite.
(b) [10 pt] Recall from the ¯rst day of class that if
p
is a prime such that 2
p
¡
1 is also prime, then we
set
M
p
= 2
p
¡
1 and call
M
p
a
Mersenne prime
. Write a careful proof of the following fact (style and
grammar counts here):
If
M
p
is a Mersenne prime, then
M
2
p
¡
1 is composite.
2.
(a) [8 pt] How many distinct anagrams of the word
ELECTRICITY
are there (including nonsensical
words)?
(b) [7 pt] How many distinct two word combinations can be made from this same word? For example,
two such combinations are
TREEL CICITY
and
L TEECITCIYR
.
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3.
De¯ne
R
=
f
(
a;b
)
2
Z
£
Z
:
a
and
b
are both even or both odd
g
:
Then
R
is a relation on
Z
.
(a) [10 pt] Of the ¯ve properties relations can have, name
all
the properties
R
has.
No proofs are
necessary here.
(b) [3 pt] Is
R
an equivalence relation?
YES
or
NO
(circle one.)
(c) [7 pt] If your answer to part (b) is
YES
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 Spring '10
 Jooe
 Natural number, Equivalence relation, Prime number

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